Motivic twisted K-theory

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Aug 29, 2010 - We thank J. Lurie for this remark. It is unclear if ...... polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volume 3 of Int. Press. Lect. Ser.
arXiv:1008.4915v1 [math.AT] 29 Aug 2010

Motivic twisted K-theory Markus Spitzweck, Paul Arne Østvær August 1, 2010 Abstract This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGm -bundle for the classifying space of the multiplicative group scheme. We show a K¨ unneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BGm . The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theory. By adopting the notion of an E∞ -ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted K-groups. It generalizes various spectral sequences computing the algebraic K-groups of schemes over fields. Moreover, we construct a Chern character between motivic twisted K-theory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.

Contents 1 Motivation

2

2 Main results

2

3 Main definitions and first properties

7

4 A K¨ unneth isomorphism for homological motivic twisted K-theory

11

5 Spectral sequences for motivic twisted K-theory

18

6 Further problems and questions

27

7 Graded Adams Hopf algebroids

28

1

1

Motivation

Topological K-theory has many variants which have all been developed and exploited for geometric purposes. Twisted K-theory or “K-theory with coefficients” was introduced by Donovan and Karoubi in [10] using Wall’s graded Brauer group. More general twistings of K-theory arise from automorphisms of its classifying space of Fredholm operators on an infinite dimensional separable complex Hilbert space. Of particular geometric interest are twistings given by integral 3-dimensional cohomology classes. The subject was further developed in the direction of analysis by Rosenberg in [37]. Twisted K-theory resurfaced in the late 1990’s with Witten’s work on classification of D-branes charges in type II string theory [52]. Fruitful interactions between algebraic topology and physics afforded by twisted K-theory continues today, see e.g. [1], [5], [6], [8] and [46]. The work of Freed, Hopkins and Teleman [14] relates the twisted equivariant K-theory of a compact Lie group G to the “Verlinde ring” of its loop group. For a recent survey of twisted K-theory we refer to [25]. We are interested in twistings of the motivic K-theory spectrum KGL in the algebrogeometric context of motivic homotopy theory [12], [47]. Over the complex numbers C, or more generally any field with a complex embedding, our motivic twisted K-theory specializes to twisted K-theory under realization of complex points. The idea of twisting (co)homology theories have been used to great effect in topology. A classical example is cohomology with local coefficients, which can be used to formulate Poincar´e duality and the Thom isomorphism for unorientable manifolds. Analogous motivic results are subject to future work. In [4], Ando, Blumberg and Gepner use the formalism of ∞-categories in order to construct twisted forms of multiplicative generalized (co)homology theories, and in [29], May and Sigurdsson employ parametrized spectra to the same end. Their setups suggest analogous algebro-geometric generalizations.

2

Main results

The isomorphism classes of principal BS 1 -bundles over a topological space X identifies canonically with the homotopy classes of maps from X to the Eilenberg-MacLane space K(Z, 3). The second delooping BBS 1 of the circle gives a concrete model for K(Z, 3). We begin the paper by considering the analogous setup in motivic homotopy theory. Let X be a motivic space and Gm the multiplicative group scheme over a noetherian base scheme S of finite dimension (usually left implicit in the notation).

2

For any map τ : X → BBGm define X τ as the homotopy pullback along ∗ → BBGm (which can be thought of as a universal principal BGm -bundle): Xτ 

X

/∗  / BBGm

τ

With this definition there is a naturally induced module map P∞ × X τ → X τ . Here we use implicitly the motivic weak equivalence between BGm and the infinite projective space P∞ . By passing to motivic suspension spectra we get a naturally induced map ∞ τ ∞ τ Σ∞ P∞ + ∧ Σ X+ → Σ X+ displaying Σ∞ X+τ as a module over the motivic ring spectrum Σ∞ P∞ + . (We defer the somewhat technical definition of this module structure to Section 3.) When S is a smooth scheme over a field we can identify the homotopy classes of maps from X to BBGm with the third integral motivic cohomology group MZ3,1 (X ) of weight one. This group is in fact trivial for smooth schemes of finite type over S by [45, Corollary 3.2.1]. On the other hand, it is nontrivial for motivic spheres. Denote by BGL the classifying space of the infinite Grassmannian over S. Suppose fζ : X → P∞ and fξ : X → Z × BGL represent ζ ∈ MZ2,1 (X ) and ξ ∈ KGL∗ (X ), respectively. Then the composite map ∆

X →X ×X

fζ ×fξ

→ P∞ × (Z × BGL) → Z × BGL

represents an element ξ⊗ζ ∈ KGL∗ (X ). The above defines the action of the Picard group on the K-theory ring of X . On the level of motivic spectra there exists a corresponding composite map Σ∞ P∞ + ∧ KGL → KGL ∧ KGL → KGL, where the second map is the ring multiplication on KGL. The first map is obtained via adjointness from the multiplicative map BGm → {1} × BGL that sends a line bundle represented by a map into P∞ to its class in the Grothendieck group of all vector bundles [43, (3)]. Thus the motivic K-theory spectrum KGL is a module over Σ∞ P∞ +. We are now ready to define our main objects of study in this paper. The distinction between homological and cohomological versions of motivic twisted K-theory is rooted in standard nomenclature for twisted K-theory. 3

Definition 2.1: For τ : X → BBGm define the motivic twisted • homological K-theory of τ by KGLτ ≡ Σ∞ X+τ ∧Σ∞ P∞ KGL. + • cohomological K-theory of τ by KGLτ ≡ HomΣ∞ P∞ (Σ∞ X+τ , KGL). + The smash product in the definition of KGLτ is derived in the sense that it is formed in the homotopy category of highly structured modules over Σ∞ P∞ + . In order to make sense of the derived smash product, we implicitly use a closed symmetric monoidal model for the motivic stable homotopy category, see Jardine’s work on motivic symmetric spectra [24], for example. Later in the paper we prove that homotopy equivalent maps from X to BBGm give rise to isomorphic motivic twisted K-theories. In addition, the derived style definition of KGLτ requires a strict ring model for KGL as a Σ∞ P∞ + -ring spectrum. Such a model was only recently constructed in [40] using the Bott tower β

−2,−1 ∞ ∞ Σ P+ Σ∞ P∞ + →Σ

Σ−2,−1 β

→

··· .

(1)

Similarly, in the cohomological setup, the hom-object appearing in the definition of KGLτ is formed in the homotopy category of Σ∞ P∞ + -modules. τ An alternate definition of KGL can be made precise by simply inverting the (2, 1) self-map of Σ∞ X+τ obtained from the Bott map realizing K-theory KGL as the Bott inverted infinite projective space. (The Bott map β is indeed a Σ∞ P∞ + -module map by construction.) Independent proofs of the latter result have appeared in [16] and [43]. For other discussions of the Bott inverted model for K-theory we refer to [31], [32], [40]. Making use of the Bott map provides also an alternate definition of KGLτ . We shall be using this viewpoint on a number of occasions in this paper. Now suppose the twisting class τ for X is null and the base scheme S is regular. In Section 3 we show that the homotopy fiber X τ identifies with the product P∞ × X and KGLτ identifies with the smash product Σ∞ X+ ∧ KGL. Accordingly, we may view motivic twisted K-theory as a generalization of K-theory. In the course of the paper we shall work out some of the differences and similarities arising from this generalization, and suggest some open problems. For a general twist τ it turns out that the motivic twisted K-theory spectra KGLτ and KGLτ do not acquire ring structures in the motivic stable homotopy category. In particular, the motivic twisted K-groups KGLτ∗ do not form a ring in general. The lack of a product structure tends to complicate computations. On the other hand we establish two far more powerful tools for performing computations. First we prove a K¨ unneth isomorphism for homological motivic twisted K-groups and second we construct spectral sequences relating motivic (co)homology to motivic twisted K-groups. 4

By applying the Tor-spectral sequence in [11, Proposition 7.7] to the commutative ∞ τ motivic ring spectrum Σ∞ P∞ + ∧ KGL and its modules KGL and Σ X+ ∧ KGL we arrive at the strongly convergent tri-graded spectral sequence KGL (P∞ )

∗ Tora,(b,c)

(KGL∗ (X τ ), KGL∗ ) ⇒ KGLτa+b,c (X ).

(2)

Here we should infer that Σ∞ P∞ + ∧KGL and KGL are stably cellular motivic ring spectra, a.k.a. “Tate spectra” [31]. To begin with, the motivic K-theory spectrum KGL is stably cellular [11, Theorem 6.2]. The suspension spectrum of the pointed infinite projective space is also stably cellular by [11, Propositions 2.13, 2.17, Lemma 3.1] since any filtered colimit of stably cellular motivic spaces is stably cellular [11, Definition 2.1(3)]. Hence the smash product Σ∞ P∞ + ∧ KGL is cellular. (Although the case of fields is emphasized in [11] the results we employ from loc. cit. hold over arbitrary noetherian base schemes of finite dimension.) Theorem 2.2: The edge homomorphism in the Tor-spectral sequence (2) induces a natural isomorphism KGLτ∗ (X ) ∼ = KGL∗ (X τ ) ⊗KGL∗ (P∞ ) KGL∗ . This is the motivic analogue of the corresponding topological result shown in [26]. KGL∗ (P∞ ) (KGL∗ , KGL∗ (X τ )) is Theorem 2.2 follows from (2) by proving the Tor-group Tor a,(b,c) trivial for a > 0. It is worthwhile to point out that KGL∗ is not a flat KGL∗ (P∞ )-module, i.e. the vanishing result for the Tor-groups does not hold for an “obvious” reason. Our proof of the vanishing employs flatness of the ring map KGL∗ (P∞ ) → KGL∗ KGL and the homotopy theory of Hopf algebroids. More precisely, it is shown that the composite map KGL∗ (P∞ ) → KGL∗ KGL → KGL∗ satisfies the Landweber exactness criterion relative to the Hopf algebroid (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )). Furthermore, KGL∗ (X τ ) is a comodule over the same Hopf algebroid, and the Tor-groups are computed by a cofibrant replacement in the projective model structure on the category of unbounded chain complexes of such comodules. A crux input is that the Hopf algebroid in question is an “Adams Hopf algebroid.” This notion is recalled in Section 7 together with some background from stable homotopy theory. Combining these facts we show the desired vanishing of the Tor-groups in positive degrees. Our use of the model structure circumvents a more explicit construction employed in the topological situation [26]. Algebraic K-theory is closely related to motivic cohomology and more classically to higher Chow groups via Chern characters. We shall briefly examine a Chern character for motivic twisted K-theory with target twisted periodized rational motivic cohomology Chτ : KGLτ → PMτ Q. 5

The construction of Chτ follows the setup for the Chern character for KGL in [32]. As it turns out, the rationalization of Chτ is an isomorphism under a mild assumption on the base scheme originating in the work of Cisinski-D´eglise [9]. (We leave the formulation of the corresponding result for KGLτ to the main body of the paper.) Theorem 2.3: For geometrically unibranched excellent base schemes the rational Chern character ChτQ : KGLτQ → PMτ Q. is an isomorphism in the homotopy category of modules over Σ∞ P∞ +. Section 4 provides streamlined proofs of the results reviewed in the above. In the same section we work out explicit computations of the motivic twisted K-groups of the motivic (3, 1)-sphere. Over finite fields the motivic twisted K-groups in positive degrees 2k − 1 and 2k turn out to be finite cyclic groups of the same order. This amusing computation is closely related to Quillen’s computation of the K-groups of finite fields. Some of the basic facts concerning flat Adams Hopf algebroids required in Section 4 are deferred to Section 7. In Section 5 we establish powerful integral relations between motivic (co)homology and motivic twisted K-theory in the form of spectral sequences MZ∗ (Σ∞ X+ ) =⇒ KGLτ∗ (X ) and MZ∗ (Σ∞ X+ ) =⇒ KGL∗τ (X ). For some closely related papers on spectral sequences computing (non-twisted) K-groups in terms of motivic cohomology groups we refer the reader to [7], [15], [17], [27], [44] and [49]. The question of strong convergence of these spectral sequences is a tricky problem. Our approach involves the very effective motivic stable homotopy category SH(S)Veff . We define it as the smallest full subcategory of the motivic stable homotopy category SH(S) that contains suspension spectra of smooth schemes of finite type over S and is closed under extensions and homotopy colimits. This is not a triangulated category; however, it is a subcategory of the effective motivic stable homotopy category SH(S)eff . (In fact, it is the homologically positive part of a t-structure on SH(S)eff .) The very effective motivic stable homotopy category is of independent interest. We show that the algebraic cobordism spectrum MGL lies in SH(S)Veff . When S is a field of characteristic zero, we show that the connective K-theory spectrum kgl lies in SH(S)Veff . This is a key input for showing strong convergence of the spectral sequences. 6

The main body of the paper ends in Section 6 with a discussion of open problems. In particular, we suggest extending the techniques in this paper to the settings of both equivariant K-theory and hermitian K-theory. For legibility, bigraded motivic homology theories are written with a single grading. The precise meaning of the gradings should always be clear from the context.

3

Main definitions and first properties

In this section we first put the definitions of the motivic twisted K-theory spectra on rigorous grounds. This part deals with model structures and classifying spaces. The constructions are rigged so that smashing with the sphere spectrum over Σ∞ P∞ + yields a useful “untwisting” result detailed in Lemma 3.6. For algebro-geometric reasons we shall explain below, some of the results require mild restrictions on the base scheme. Let Spc be the category of motivic spaces on Sm, i.e. simplicial presheaves on the Nisnevich site of smooth schemes of finite type over S, with the injective motivic model structure. This model structure satisfies the monoid axiom [41]. Hence for any monoid G in Spc, the category Mod(G) of G-modules acquires a model structure. For a map G → H of monoids there is an induced left Quillen functor Mod(G) → Mod(H). In particular, the pushforward of a G-module X along the canonical map G → ∗ is the quotient X /G. The homotopy quotient is defined similarly by first taking a cofibrant replacement of X in Mod(G). We denote by ModY (G) the category of G-modules in the slice category Spc/Y comprised of motivic spaces over a motivic space Y . It should be noted that ModY (G) is a model category: To begin with, Spc/Y inherits an evident model structure from the motivic model structure on Spc which turns the pairing Spc × Spc/Y → Spc/Y sending (X , X ′ → Y ) to (X × X ′ ) → X ′ → Y into a Quillen bifunctor. Moreover, every object of Spc is cofibrant. Thus the existence of the model structure on ModY (G) follows from a relative version of [41, Theorem 3.1.1]. For a G-module X and a map of motivic spaces Y ′ → Y , note that X ∈ ModX /G (G) and there is a pullback functor ModY (G) → ModY ′ (G). In what follows we specialize to the commutative monoid BGm ≃ P∞ . As a model for the classifying space BP∞ of P∞ we may use the standard bar construction. Viewing ∗ as a P∞ -module we are entitled to a cofibrant replacement Q → ∗ in Mod(P∞ ). The homotopy quotient Q/P∞ gives an alternative model for the classifying space of P∞ . In the proof of Lemma 3.6 we find it convenient to use the latter.

7

If τ : X → Q/P∞ is a fibration in Spc, then the homotopy pullback X τ ≡ X ×Q/P∞ Q ∈ ModX (P∞ ) is a P∞ -module over X , in particular a P∞ -module. Suppose τ : X → (Q/P∞ )f is a map in Spc with target some fibrant replacement of Q/P∞ . The model structure ensures there exists a functorial fibrant replacement (Q)f of Q in Mod(Q/P∞ )f (P∞ ). Using these fibrant replacements we define X τ by the homotopy pullback X τ ≡ X ×(Q/P∞ )f (Q)f ∈ ModX (P∞ ). Working in motivic symmetric spectra we note that Σ∞ X+τ is a strict Σ∞ P∞ + -module. (Recall that Σ∞ is a left Quillen functor for the injective motivic model structure [24].) The motivic twisted homological K-theory of τ : X → (Q/P∞ )f in Spc is defined as the derived smash product KGL. KGLτ ≡ Σ∞ X+τ ∧Σ∞ P∞ + For the strict module structure on KGL we use the Bott inverted model discussed in [40]. Likewise, by running the same fibrant replacements, the motivic twisted cohomological K-theory of τ : X → (Q/P∞ )f is defined as the derived internal hom (Σ∞ X+τ , KGL). KGLτ ≡ HomΣ∞ P∞ + In the following we let BBGm denote the homotopy quotient (Q/P∞ )f . Proposition 3.1: Suppose S is a regular scheme and τ : X → BBGm . If τ is null then KGLτ is isomorphic to Σ∞ X+ ∧ KGL in the motivic stable homotopy category. Proof. Corollary 3.4 identifies X+τ with the smash product of motivic pointed spaces X+ ∧ P∞ + . This follows because of the assumption on τ the former is obtained by first taking the homotopy pullback of the diagram ∗ → BBGm ← ∗ and second forming the homotopy pullback along the canonical map X → ∗. Using this we obtain the isomorphisms Σ∞ X+τ ∧Σ∞ P∞ KGL ∼ KGL ∼ = (Σ∞ X+ ∧ Σ∞ P∞ = Σ∞ X+ ∧ KGL. + ) ∧Σ∞ P∞ + + The regularity assumption on the base scheme S enters the proof of Corollary 3.4, which we discuss next. 8

Recall from [31, §2] the definition of the simplicial Picard functor νPic on Sm: For a scheme X in Sm, let Pic(X) denote the associated Picard groupoid. Then the pseudofunctor X 7→ Pic(X) can be strictified to a presheaf on Sm. Applying the nerve functor to any such strictification defines the simplicial presheaf νPic on Sm. Lemma 3.2: Suppose S is a normal scheme. Then the simplicial Picard functor νPic is a Nisnevich local A1 -invariant simplicial presheaf. Proof. Nisnevich localness holds because the groupoid valued Picard functor Pic satisfies flat descent. And A1 -invariance holds because Pic is A1 -invariant by assumption. We remark that νPic is a commutative monoid in Spc by strictification. Lemma 3.3: Suppose S is a regular scheme. Applying the classifying space functor sectionwise to νPic determines a Nisnevich local A1 -invariant simplicial presheaf. 2 (X, G ) Proof. Since S is regular, it is well known that the cohomology group HNis m is trivial for every X in Sm. For an outline of a proof, we note that the sheaf MX∗ of meromorphic functions on X and the sheaf ZX1 of codimension 1 cycles on X are flasque 2 (X, G ) follows in the Nisnevich topology. Thus, using [18, §21.6], the vanishing of HNis m from the exact sequence ∗ → MX∗ → ZX1 → 0. (3) 0 → OX

Let B s νPic be the sectionwise classifying space of νPic and ϕ : B s νPic → RB s νPic a 2−i Nisnevich local replacement. Then πi ((RB s νPic)(X)) = HNis (X, Gm ) for 0 ≤ i ≤ 2. It follows that ϕ is a sectionwise equivalence. Thus the sectionwise classifying space construction is Nisnevich local (sectionwise equivalent to every Nisnevich local fibrant replacement) and A1 -invariance is preserved. Corollary 3.4: If S is a regular scheme then the homotopy pullback of the diagram ∗ → BBGm ← ∗ is isomorphic to P∞ in the motivic homotopy category. Proof. Let Hs (S) denote the homotopy category of simplicial presheaves on Sm with the objectwise model structure, and let H(S) denote the motivic homotopy category. Then the inclusion H(S) → Hs (S) arises from a right Quillen functor. Therefore, in order to compute the homotopy pullback of ∗ → BBGm ← ∗ in the motivic homotopy category, it is sufficient to compute the homotopy pullback of its image in Hs (S). 9

By Lemma 3.3, B s νPic is an A1 - and Nisnevich local replacement of BBGm (for notation see the proof of Lemma 3.3). The homotopy pullback of ∗ → B s νPic ← ∗ in Hs (S) is clearly BGm . Remark 3.5: The previous corollary would have been trivially true provided the infinity category of motivic spaces had been an infinity topos in the sense of [28]. Alas, this is not true for motivic spaces: Recall that in any infinity topos the loop functor provides an equivalence between the connected 1-truncated pointed objects and discrete group objects. In motivic homotopy theory, the simplicial loop space of a 1-truncated pointed motivic space is strongly A1 -invariant. Recall also that A1 -invariant and strongly A1 -invariant sheaves of groups are different notions. A discrete motivic group object is synonymous with an A1 -invariant sheaf of groups. This shows that the equivalence does not hold in the motivic setting. We thank J. Lurie for this remark. It is unclear if Corollary 3.4 extends to an interesting class of base schemes. We note that the sequence (3) is exact if and only if X is a locally factorial scheme. However, a smooth scheme of finite type over a locally factorial scheme need not be locally factorial in general. Thus we cannot 2 (X, G ) is trivial over every locally factorial base scheme. expect that the group HNis m We thank M. Levine for clarifying this remark. ∞ τ 1∼ Lemma 3.6: There is an isomorphism of Σ∞ P∞ = Σ ∞ X+ + -modules Σ X+ ∧Σ∞ P∞ + ∞ where Σ∞ P∞ + → 1 is induced by the canonical map P → ∗.

Proof. For a fibration Y ′ → Y in Spc the pullback functor Spc/Y → Spc/Y ′ is a left Quillen functor for the injective motivic model structure on Spc. Indeed, it has a right adjoint and it preserves monomorphisms and weak equivalences. (Recall that Spc is right proper.) Thus we obtain a left Quillen functor ModY (P∞ ) → ModY ′ (P∞ ). This functor commutes with pushforward along the canonical map P∞ → ∗ and thus it preserves homotopy quotients by P∞ . We deduce that the natural map QX τ /P∞ → X is a weak equivalence, where QX τ → X τ is a cofibrant replacement in ModX (P∞ ). On the other hand, the forgetful functor ModX (P∞ ) → Mod(P∞ ) is also a left Quillen functor. Combining the above findings shows there is an isomorphism ∼ X τ ×L P∞ ∗ = X . Applying the motivic symmetric suspension spectrum functor yields the result.

10

Next we consider some basic functorial properties of motivic twisted K-theory. First we note there exits a functor KGL− : Ho(Spc/BBGm ) → SH(S). Note here that for a map from τ : X → BBGm to τ ′ : X ′ → BBGm there is an induced ′ map between pullbacks X τ → (X ′ )τ of P∞ -modules, which induces a map of motivic ′ symmetric spectra KGLτ → KGLτ . Clearly this factors through the homotopy category of motivic spaces over BBGm . In particular, if τ and τ ′ are A1 -homotopy equivalent ′ maps, then KGLτ and KGLτ are isomorphic. We also note that KGL− can be enhanced to a functor from Ho(Spc/BBGm ) taking values in the category of highly structured KGL-modules. Likewise, in the cohomological setup, there exists a functor KGL− : Ho(Spc/BBGm )op → SH(S). Some parts of our discussion of motivic twisted K-theory rely on the notion of a “motivic E∞ -ring spectrum.” For every operad O in motivic symmetric spectra, the category of O-algebras acquires a combinatorial model structure. A recent account of this model structure has been written up by Hornbostel in [20]. Here, motivic symmetric spectra are viewed in the stable flat positive model structure. In particular, there exists model structures for the commutative motivic operad and the image of the linear isometry operad in topological spaces. A motivic E∞ -ring spectrum is an algebra over a Σ-cofibrant replacement of the commutative motivic operad. Motivic E∞ -ring spectra and strict commutative motivic ring spectra are related by a Quillen equivalence [20]. For our purposes we may therefore use these two notions interchangeably.

4

A K¨ unneth isomorphism for homological motivic twisted K-theory

An explicit computation furnishes a natural base change isomorphism expressing the KGL-homology of P∞ in terms of KGL∗ and unitary topological K-theory KGL∗ (P∞ ) ∼ = KGL∗ ⊗KU∗ KU∗ (CP∞ ).

(4)

The multiplicative structure on KGL∗ (P∞ ) induced from the H-space structure on P∞ can be read off from this isomorphism by using the ring structure on KU∗ (CP∞ ) and the coefficient ring. We refer to the work of Ravenel and Wilson [36] for a description of the ring structure on KU∗ (CP∞ ) in terms of the multiplicative formal group law. 11

An application of motivic Landweber exactness [31, Proposition 9.1] shows there is a natural base change isomorphism KGL∗ KGL ∼ = KGL∗ ⊗KU∗ KU∗ KU.

(5)

The multiplicative structure on KGL∗ KGL induced from the ring structure on KGL can be read off from this isomorphism by using the ring structure on KU∗ KU and the coefficient ring. For a description of the Hopf algebra KU∗ KU we refer to the work of Adams and Harris [2, Part II, §13]. Lemma 4.1: Under the naturally induced composite map KGL∗ (P∞ ) → KGL∗ KGL → KGL∗ the generator βi maps to 1 if i = 0, 1 and to 0 if i 6= 0, 1. Proof. We note that KGL∗ (P∞ ) is a free KGL∗ -module generated by the elements βi for i ≥ 0. Thus the claim follows from the analogous result for unitary topological K-theory of CP∞ , see [26] for example, by applying the functor KGL∗ ⊗KU∗ − and appealing to the base change isomorphisms (4) and (5). Lemma 4.1 verifies the previous assertion that KGL∗ is not a flat KGL∗ (P∞ )-module. Next we establish a result which is pivotal for our proof of the vanishing of the Tor-groups discussed in Section 2. Lemma 4.2: The naturally induced map KGL∗ (P∞ ) → KGL∗ KGL is a flat ring map. Proof. In [16] and [43] it is shown that KGL is isomorphic to the Bott inverted motivic suspension spectrum of P∞ + . Thus the map in question is a localization. In particular it is flat. For an alternate proof, combine the base change isomorphisms (4) and (5) with flatness of the naturally induced map KU∗ (CP∞ ) → KU∗ KU. (This map is a localization according to the topological analogue of our first argument, which is well known and follows from the motivic result by taking complex points, or alternatively by [26].) In Proposition 7.3 we show that (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) has the structure of a flat graded Adams Hopf algebroid. We refer the reader to Section 7 for background on the notions and results appearing in the formulation and proof of the following key result.

12

Theorem 4.3: The naturally induced composite map KGL∗ (P∞ ) → KGL∗ KGL → KGL∗ is Landweber exact for the flat graded Adams Hopf algebroid (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )). Proof. By Lemma 7.6 it suffices to show that the left unit map ηKGL∗ (P∞ ) : KGL∗ (P∞ ) → KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ ) ∼ = (KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) ⊗KGL∗ (P∞ ) KGL∗ (P∞ ) → (KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) ⊗KGL∗ (P∞ ) KGL∗ for the displayed Hopf algebroid is flat (the target is canonically isomorphic to KGL∗ KGL). Remark 7.4 provides more details on the Hopf algebroid structure. The left unit map ηKGL∗ (P∞ ) and its topological analogue ηKU∗ (CP∞ ) determines a commutative diagram where the horizontal maps are the base change isomorphisms given in (4) and (5): KGL∗ (P∞ )

∼ =

/ KGL∗ ⊗KU KU∗ (CP∞ ) ∗

ηKGL∗ (P∞ )

KGL∗ ⊗ηKU∗ (CP∞ )



∼ =

KGL∗ KGL



/ KGL∗ ⊗KU KU∗ KU ∗

Khorami [26] has shown that ηKU∗ (CP∞ ) coincides with the naturally induced map from KU∗ (CP∞ ) to KU∗ KU. By motivic Landweber exactness [31] we deduce that ηKGL∗ (P∞ ) coincides with the naturally induced flat map in Lemma 4.2. This finishes the proof. By Landweber exactness the functor from comodules over KGL∗ KGL⊗KGL∗ KGL∗ (P∞ ) to KGL∗ -algebras is exact (where KGL∗ is viewed with its KGL∗ (P∞ )-algebra structure). This observation is a crux input in the proof of the next result. Corollary 4.4: Suppose E is a comodule over KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ ). The group KGL∗ (P∞ )

Tor ∗

(E, KGL∗ )

is trivial in positive degrees.

13

Proof. Proposition 7.3 implies there exists a projective model structure on the category of non-connective chain complexes of KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )-comodules for the set of dualizable comodules [21, Theorem 2.1.3]. The projective model structure is proper, finitely generated, stable symmetric monoidal and satisfies the monoid axiom. Moreover, a map is a cofibration if and only if it is a degreewise split monomorphism with cofibrant cokernel. The cofibrant objects are retracts of certain sequential cell-complexes described in details in [21, Theorem 2.1.3]. These results are easily transferred to the graded setting. Due to the existence of the projective model structure we are entitled to a cofibrant replacement QE → E (recall this is a projective weak equivalence and QE is cofibrant). Proposition 7.3 shows that (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) is a graded Adams Hopf algebroid. This additional structure guarantees that every weak equivalence in the projective model structure is a quasi-isomorphism [21, Proposition 3.3.1]. We claim that the Tor-groups in question are computed by the homology of the chain complex QE ⊗KGL∗ (P∞ ) KGL∗ . The proof proceeds by comparing chain complexes of comodules over the tensor product KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ ) with chain complexes of KGL∗ (P∞ )-modules. Indeed, by [21, Proposition 1.3.4], cf. the proof of [21, Theorem 2.1.3], the generating cofibrations are of such a form that QE is even cofibrant as a complex of KGL∗ (P∞ )-modules for the usual projective model structure. (We also note that the tensor factor KGL∗ need not be cofibrantly replaced because the monoid axiom holds in the projective model structure.) As noted earlier there exists an exact functor from the category of comodules over KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ ) to KGL∗ -algebras. We note that any such functor preserves quasi-isomorphisms. In particular there is a quasi-isomorphism QE ⊗KGL∗ (P∞ ) KGL∗ ≃ E ⊗KGL∗ (P∞ ) KGL∗ . By combining the above we conclude that the Tor-groups vanish in positive degrees. By strong convergence of the spectral sequence (2) we are almost ready to conclude the proof of the K¨ unneth isomorphism in Theorem 2.2. It only remains to observe that τ KGL∗ (X ) is a comodule over the Hopf algebroid (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )). To begin with, the naturally induced action of P∞ on X τ yields a map KGL∗ (P∞ × X τ ) → KGL∗ (X τ ). 14

Since KGL∗ (P∞ ) is free over the coefficient ring KGL∗ there is an isomorphism KGL∗ (P∞ × X τ ) ∼ = KGL∗ (P∞ ) ⊗KGL∗ KGL∗ (X τ ). It follows that KGL∗ (X τ ) is a module over KGL∗ (P∞ ). Using the unit map from the motivic sphere spectrum 1 to KGL we get a map between motivic spectra KGL ∧ Σ∞ X+τ ∼ = KGL ∧ 1 ∧ Σ∞ X+τ → KGL ∧ KGL ∧ Σ∞ X+τ . From this we immediately obtain the desired comodule map KGL∗ (X τ ) → KGL∗ KGL ⊗KGL∗ KGL∗ (X τ ) ∼ = (KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) ⊗KGL∗ (P∞ ) KGL∗ (X τ ). This is clearly a coassociative and unital map between KGL∗ (P∞ )-modules. Remark 4.5: As noted earlier the KGL∗ -module KGL∗ (P∞ ) is free on the generators βi for i ≥ 0. Its multiplicative structure can be described in terms of power series. Modulo the problem of computing the coefficient ring KGL∗ this leaves us with investigating the K-theory of the homotopy fiber X τ . On the other hand, an inspection of the module structures in Theorem 2.2, cf. Lemma 4.1, reveals there is an isomorphism KGLτ∗ (X ) ∼ = KGL∗ (X τ )/(β0 − 1, β1 − 1, βi )i≥2 . In case τ is the identity map on BBGm then KGL∗ (BBGτm ) ∼ = KGL∗ . We claim that KGLτ∗ (BBGm ) is the trivial group. This follows by comparing the images of β0 or β1 in the respective tensor factors. For example, the class β1 maps to the unit in KGL∗ and to zero in KGL∗ (BBGτm ). Next we turn to the constructions of the twisted Chern characters. The proof of our main result Theorem 2.3 relies on results in [9] and [32]. Let MQ denote the motivic Eilenberg-MacLane spectrum introduced in [47]. (We refer to [13] for a definition of MQ viewed as a motivic functor.) It has the structure of a commutative monoid in the category of motivic symmetric spectra [38], [39]. The periodization PMQ of MQ is also highly structured: Form the free commutative MQ-algebra PMQ≥0 on one generator in degree (2, 1) (perform this in P1 -spectra of simplicial presheaves of Q-vector spaces and then transfer the spectrum back to obtain a strictly commutative ring spectrum). Inverting the same generator following the method in [40] produces the commutative W monoid PMQ whose underlying spectrum is the infinite wedge sum i∈Z Σ2i,i MQ. 15

Lemma 4.6: There is an E∞ -isomorphism between PMQ and KGL ∧ MQ. Proof. By the universal property of PMQ≥0 there is a commutative diagram: Σ∞ P∞ + ∧ MQ

oo7 ooo o o o ooo  / KGL ∧ MQ PMQ≥0

The generator in degree (2, 1) maps to the canonical element in Σ∞ P∞ + ∧MQ determined by the Bott element of Σ∞ P∞ [43]. The diagonal map is an isomorphism. Inverting the + generator and the Bott element gives the desired isomorphism. Lemma 4.6 furnishes an Σ∞ P∞ + -algebra structure on PMQ via the map ∼ Σ∞ P∞ + → KGL → KGL ∧ MQ = PMQ. Combining Lemma 4.6 and the canonical ring map KGL → KGL ∧ MQ we arrive at the Chern character Ch : KGL → PMQ (6) from algebraic K-theory to the periodized rational motivic Eilenberg-MacLane spectrum. (This is a map of motivic ring spectra.) For any twist τ , smashing (6) with X τ in the homotopy category of Σ∞ P∞ + -modules defines the twisted Chern character Chτ : KGLτ → PMτ Q.

(7)

As asserted in Theorem 2.3, the rationalization of (7) is an isomorphism for geometrically unibranched excellent base schemes. This follows by combining [9, Corollary 15.1.6] and [32, Theorem 10.1, Corollary 10.3]. Similarly we define the cohomological Chern character Chτ : KGLτ → PMτ Q

(8)

by taking internal hom-objects from Σ∞ X+τ into the untwisted Chern character Ch. We note that the rationalization of (8) is an isomorphism over geometrically unibranched excellent base schemes provided Σ∞ X+τ is strongly dualizable in Ho(Σ∞ P∞ + − Mod). Indeed, this follows immediately by smashing the rational isomorphism in (6) with the dual of Σ∞ X+τ .

16

Remark 4.7: In the topological setup, Atiyah and Segal [6] employed a different method in order to construct a Chern character for twisted K-theory and a corresponding theory of Chern classes. We leave the comparison of the two constructions as an open question. We end this section by outlining computations of nontrivial twisted K-groups for the motivic (3, 1)-sphere. To begin with we allow the base scheme to be an arbitrary field. In the interest of explicit computations in all degrees, we specialize to finite fields. Recall the smash product decomposition S 3,1 = S 2 ∧Gm for the motivic (3, 1)-sphere. Moreover, there is a homotopy pushout square of motivic spaces: P1

/∗



 / S 3,1

∗

We shall consider the twist τn : S 3,1 → BBGm corresponding to n times the canonical map S 3,1 → BBGm . Precomposing with the map ∗ → S 3,1 produce null homotopic twists on P1 and the point. In order to proceed we infer, leaving details to the interested reader, there is a homotopy pushout diagram: P∞ × P1

/ P∞ × (P∞ )n

/ P∞  / (S 3,1 )τn



P∞

The left vertical map is the projection on the first factor. The upper composite horizontal map arise from embedding P1 into (P∞ )n along the diagonal map P∞ ⊆ (P∞ )n and using the H-space structure on the infinite projective space. With this in hand we get an induced long exact sequence · · · → Σ2,1 KGL∗ ⊕ KGL∗ → KGL∗ ⊕ KGL∗ → KGLτ∗n (S 3,1 ) → · · · .

(9)

Next we infer that the map between the direct sums in (9) is uniformly given by (a, b) 7→ (anβ + b, −b).

(10)

Again we leave the details to the interested reader. (Note that (10) is compatible with its evident topological counterpart.) From (9) we deduce the exact sequence K1 ⊕ K1 → K1 ⊕ K1 → K1τn (S 3,1 ) → K0 ⊕ K0 → K0 ⊕ K0 → K0τn (S 3,1 ) → 0. 17

(11)

Using (11) and the fact that K0 is infinite cyclic for any field we read off the isomorphism K0τn (S 3,1 ) ∼ = Z/n, where, in general, Kiτ (X ) is shorthand for KGLτi,0 (X ), i ∈ Z. By specializing to a finite field Fq and an odd integer i ≥ 1, we deduce the exact sequence τn 0 → Ki+1 (SF3,1 ) → Ki ⊕ Ki → Ki ⊕ Ki → Kiτn (SF3,1 ) → 0. q q

(12)

This follows from (9) since the K-groups for finite fields vanish in positive even degrees [34]. Combining (10) and (12) yields the isomorphisms ×n τn K2i (SF3,1 )∼ = ker(Z/(q i − 1) −→ Z/(q i − 1)) ∼ = Z/gcd(n, q i − 1) q

and τn K2i−1 (SF3,1 )∼ = Z/gcd(n, q i − 1). q

5

Spectral sequences for motivic twisted K-theory

In this section we shall construct and show strong convergence of the spectral sequences relating motivic (co)homology to motivic twisted K-theory. The review of this material in Section 2 provides motivation and background from K-theory. Our approach employs the slice tower formalism introduced by Voevodsky [48] and further developed from the viewpoint of colored operads in [19]. Let ri denote the right adjoint of the natural inclusion functor ΣiT SH(S)eff ⊆ SH(S). Define fi : SH(S) → SH(S) as the composite functor r

SH(S) →i ΣiT SH(S)eff ⊆ SH(S). In [48] the ith slice si (X) of X is defined as the cofiber of the canonical map fi+1 (X) → fi (X). In the companion paper [19] we show that f0 and s0 respect motivic E∞ -structures, and fq and sq respect module structures over E∞ -algebras. Recall that f0 is reminiscent of the connective cover in topology. As a sample result we state the following key result. Theorem 5.1: Suppose A is an A∞ - or an E∞ -algebra in SptΣ T (S). Then f0 (A) is naturally equipped with the structure of an A∞ - resp. E∞ -algebra. The canonical map f0 A → A can be modelled as a map of A∞ - resp. E∞ -algebras. 18

The corresponding statements dealing with s0 and modules are formulated in [19]. In the interest of keeping this paper concise we refer to loc. cit. for further details. We define the connective K-theory spectrum kgl to be f0 KGL. With this definition, ∞ ∞ kgl is a Σ∞ P∞ + -module because the E∞ -map Σ P+ → KGL factors uniquely through the connective K-theory spectrum. Here we use that f0 is a lax monoidal functor that respects E∞ -objects. More generally, fi KGL = Σ2i,i kgl is a Σ∞ P∞ + -module. (The two possible module structures, using either the shift functor or the fact that fi produces a module over f0 , coincide.) Moreover, fi+1 KGL → fi KGL is a kgl-module map, hence a Σ∞ P∞ + -module map. By stitching these maps together we obtain a sequential filtration of KGL by shifted copies of the connective K-theory spectrum · · · → Σ2i+2,i+1 kgl → Σ2i,i kgl → · · · → KGL.

(13)

The maps in (13) are Σ∞ P∞ + -module maps. Hence for every twist τ : X → BBGm there is an induced filtration of the motivic twisted K-theory spectrum · · · → Σ∞ X+τ ∧Σ∞ P∞ Σ2i+2,i+1 kgl → Σ∞ X+τ ∧Σ∞ P∞ Σ2i,i kgl → · · · + +

(14)

KGL = KGLτ . → · · · → Σ∞ X+τ ∧Σ∞ P∞ + Likewise, by applying the functor HomΣ∞ P∞ (Σ∞ X+τ , −) to the filtration (13) of KGL + we obtain a filtration of KGLτ taking the form · · · → HomΣ∞ P∞ (Σ∞ X+τ , Σ2i+2,i+1 kgl) → HomΣ∞ P∞ (Σ∞ X+τ , Σ2i,i kgl) → · · · + +

(15)

→ · · · → HomΣ∞ P∞ (Σ∞ X+τ , KGL) = KGLτ . + Our next objective is to identify the filtration quotients Qi (X τ ) of the tower (14) and Qi (X τ ) of the tower (15). Note that the tower (14) gives rise to an exact couple by applying homotopy groups for a fixed weight: π∗ Σ∞ X+τ ∧Σ∞ P∞ Σ2i+2,i+1 kgl +

jTTTT TTTT TTTT TTTT

/ π∗ Σ∞ X+τ ∧Σ∞ P∞ Σ2i,i kgl + kkk k k kkk kkk ku kk

π∗ Qi (X τ )

Similarly, the tower (15) gives rise to an exact couple featuring the quotients Qi (X τ ). Following a standard process we obtain spectral sequences with target graded groups KGLτ∗ and KGL∗τ . In the following we analyze these spectral sequences in details when the base scheme is a perfect field. 19

From now on we assume that the base scheme S is a perfect field. Using the slice computations of KGL in [27] and [49], [50], there is an exact triangle of Σ∞ P∞ + -modules Σ2,1 kgl → kgl → MZ → Σ3,1 kgl. Thus the filtration quotient Qi (X τ ) is isomorphic to Σ∞ X+τ ∧Σ∞ P∞ Σ2i,i MZ, + whereas the filtration quotient Qi (X τ ) is isomorphic to HomΣ∞ P∞ (Σ∞ X+τ , MZ). + Lemma 5.2: The unit map 1 → Σ∞ P∞ + induces an isomorphism on zero slices. Proof. Induction on the cofiber sequence n−1 → Σ∞ Pn+ → Σ2n,n 1 Σ∞ P+

gives the isomorphism s0 1 ∼ = s0 Σ∞ Pn+ . To conclude we use that s0 commutes with homotopy colimits, cf. [42, Lemma 4.4]. Lemma 5.3: The diagram of E∞ -ring spectra Σ∞ P∞ +

/ MZ / kgl HH {= { HH {{ HH { { HH H# {{{

1

commutes. Proof. By Lemma 5.2 and the isomorphism s0 1 ∼ = MZ, applying the zero slice functor ∞ ∞ ∞ ∞ to the maps Σ P+ → kgl and Σ P+ → 1 produces diagrams of E∞ -ring spectra: Σ∞ P∞ +

/ kgl

Σ∞ P∞ +

/1



 / MZ

MZ



 / MZ

MZ

Now, since the construction of E∞ -structures on zero slices in [19] is not transparently functorial, a trick is required in order to verify commutativity of the two diagrams. This follows by applying the localization machinery of [19] to the two-colored operad whose algebras comprise maps between E∞ -algebras. 20

Theorem 5.4: There exists an isomorphism in the motivic stable homotopy category between the filtration quotient Qi (X τ ) of (14) and the (2i, i)-suspension of the motive MZ∧Σ∞ X+ of X . Likewise, there exists an isomorphism between the filtration quotient Qi (X τ ) of (15) and the (2i, i)-suspension of the internal hom Hom(Σ∞ X+ , MZ). Proof. It suffices to consider the case i = 0. The 0th filtration quotient Q0 (X τ ) identifies with Σ∞ X+τ ∧Σ∞ P∞ MZ. By Lemma 5.3 there is an isomorphism + Σ∞ X+τ ∧Σ∞ P∞ MZ ∼ 1) ∧1 MZ. = (Σ∞ X+τ ∧Σ∞ P∞ + + Lemma 3.6 implies there is an isomorphism 1) ∧1 MZ ∼ (Σ∞ X+τ ∧Σ∞ P∞ = Σ∞ X+ ∧ MZ. + The proof of the statement for Qi (X τ ) proceeds similarly by comparing the module categories over Σ∞ P∞ + and MZ via the isomorphisms Q0 (X τ ) ∼ (Σ∞ X+τ , MZ) ∼ MZ, MZ) = HomΣ∞ P∞ = HomMZ (Σ∞ X+τ ∧Σ∞ P∞ + + ∼ = Hom(Σ∞ X+ , MZ). = HomMZ (Σ∞ X+ ∧ MZ, MZ) ∼

The isomorphisms in Theorem 5.4 are clearly functorial in X and τ . It is important to note that the filtration quotients Qi (X τ ) and Qi (X τ ) are independent of the twist. Theorem 5.4 implies there exist spectral sequences MZ∗ (Σ∞ X+ ) =⇒ KGLτ∗ (X )

(16)

MZ∗ (Σ∞ X+ ) =⇒ KGL∗τ (X )

(17)

and relating motivic homology and cohomology to motivic twisted K-theory. In what follows we shall discuss the convergence properties of (16) and (17). Our approach makes use of the notion of “very effectiveness” which is of independent interest in motivic homotopy theory over any base scheme S. In order to make this precise we introduce the following subcategory of SH(S). Definition 5.5: The very effective motivic stable homotopy category SH(S)Veff is the smallest full subcategory of SH(S) that contains all suspension spectra of smooth schemes of finite type over S and is closed under extensions and homotopy colimits. 21

We note that SH(S)Veff is not a triangulated category since it is not closed under simplicial desuspension. However, it is a subcategory of the effective motivic stable homotopy category, which we denote by SH(S)eff . Finally, we remark that SH(S)Veff forms the homologically positive part of t-structures on SH(S) and SH(S)eff . Lemma 5.6: The subcategory SH(S)Veff of SH(S) is closed under the smash product. Proof. To begin with, suppose E ∈ SH(S)Veff and X ∈ Sm. Then Σ∞ X+ ∧ E lies in SH(S)Veff by the following “induction” argument on the form of E. It clearly holds when E = Σ∞ Y+ for some Y ∈ Sm. Suppose E = hocolim Ei and Σ∞ X+ ∧ Ei ∈ SH(S)Veff . Then Σ∞ X+ ∧ E ∈ SH(S)Veff because SH(S)Veff is closed under homotopy colimits. Furthermore, if in a triangle A → E → B → A[1], Σ∞ X+ ∧ A ∈ SH(S)Veff and likewise for B, then Σ∞ X+ ∧ E ∈ SH(S)Veff because SH(S)Veff is closed under extensions by definition. A similar “induction” argument in the first variable shows now that for all objects F, E ∈ SH(S)Veff the smash product F ∧ E ∈ SH(S)Veff . For the definition of the algebraic cobordism spectrum MGL we refer to [47]. One of the reasons why the category SH(S)Veff is of interest is that it contains MGL for general base schemes. Theorem 5.7: The algebraic cobordism spectrum MGL is very effective. In fact our proof of Theorem 5.7 shows the following stronger statement: The cofiber ∆ of the unit map 1 → MGL is contained in ΣT SH(S)∆ ≥0 , where SH(S)≥0 is the smallest full saturated subcategory of SH(S) that contains the suspension spectra Σ2i,i 1 for every i ≥ 0 and is closed under homotopy colimits and extensions. The notation ΣT refers to suspension with respect to the Tate object, i.e. ΣT = Σ2,1 in the usual bigrading. Lemma 5.8: Let r be an integer and suppose Σ2r,r 1 → A → B → Σ2r+1,r 1 and A → E → F → A[1] ∆ 2r,r 1 → E lies in are triangles in SH(S). If B, F ∈ Σr+1 T SH(S)≥0 then the cofiber of Σ ∆ Σr+1 T SH(S)≥0 .

22

Proof. This follows since the cofiber of Σ2r,r 1 → E is an extension of F by B and the ∆ category Σr+1 T SH(S)≥0 is closed under extensions. Let G(n, d) denote the Grassmannian parametrizing locally free quotients of rank d of the trivial bundle of rank n. Recall there is a universal subsheaf Kn,d of On and a natural map ι : G(n, d) → G(n + 1, d) that classifies the subbundle Kn,d ⊕ O of On+1 . Denote by ι the canonical point of G(n, d) obtained by the composite map ι

ι

ι

∗∼ = G(d, d) → G(d + 1, d) → · · · → G(n, d). We are interested in vector bundles of a particular type over Grassmannians. Proposition 5.9: Suppose E is a vector bundle of rank r over the Grassmannian G(n, d) ′ and O. Then ι∗ E is canonically which is a finite sum of copies of Kn,d and its dual Kn,d trivialized. Furthermore the cofiber of the map between the suspension spectra of Thom ∆ spaces Σ2r,r 1 → Σ∞ Th(E) lies in Σr+1 T SH(S)≥0 . Proof. We outline an argument which is reminiscent of the one for [42, Proposition 3.6]. The first step of the proof consists of showing there is an exact triangle ′ Σ∞ Th(ι∗ E) → Σ∞ Th(E) → Σ∞ Th(EG(n,d) ⊕ Kn,d ) → Σ∞ Th(ι∗ E)[1]

for the canonical map ι : G(n, d + 1) → G(n + 1, d + 1) (that classifies the subbundle Kn,d+1 ⊕ O ⊆ On+1 ). By induction we deduce that the cofiber of the canonical map ∆ Σ2r,r 1 → Σ∞ Th(ι∗ E) lies in Σr+1 T SH(S)≥0 . Again by applying induction, it follows that ′ ) ∈ Σr+j SH(S)∆ , where j = n − d > 0. The proposition follows Σ∞ Th(EG(n,d) ⊕ Kn,d ≥0 T now from Lemma 5.8. Next we give a proof of Theorem 5.7. Proof. We denote by ξn = colimd Kn+d,d the universal vector bundle over the infinite Grassmannian BGLn = colimd G(n + 1, d), and write MGL = hocolimn Σ−2n,−n Σ∞ Th(ξn ) = hocolimn,d Σ−2n,−n Σ∞ Th(Kn+d,d ). The unit map 1 → MGL is in turn induced by the maps Σ−2n,−n Σ∞ Th(ι∗ Kn+d,d ) → Σ−2n,−n Σ∞ Th(Kn+d,d ). By Proposition 5.9 the cofibers of these maps are contained in ΣT SH(S)∆ ≥0 . Since cofiber sequences are compatible with homotopy colimits, this finishes the proof. 23

Lemma 5.10: Let E ∈ SH(S)Veff and suppose S is the spectrum of a perfect field. Then the homotopy group πp,q (E) = 0 for p < q. Proof. For suspension spectra of smooth projective schemes of finite type the claimed vanishing is stated in [30, §5.3]. Suppose E = hocolim Ei where every Ei satisfies the conclusion of the lemma. Minor variations of [28, Corollary 4.4.2.4, Proposition 4.4.2.6] allows us to assuming the homotopy colimit is either a coproduct or a homotopy pushout. For coproducts the result is clear, while for homotopy colimits the corresponding long exact sequence of homotopy sheaves implies the vanishing. For a general extension A → E → B → A[1], where the vanishing holds for A and B, the corresponding long exact sequence of homotopy groups implies the result. We denote by SH(S)proj the full thick subcategory of SH(S) generated by the objects ΣiT Σ∞ X+ for X ∈ Sm a projective scheme and i ∈ Z. Proposition 5.11: Suppose the base scheme S is a perfect field. Let · · · → Ei+1 → Ei → Ei−1 → · · · → E be a tower of motivic spectra such that hocolim Ei = E and denote the corresponding filtration quotients by Qi . Suppose that Ei ∈ ΣiT SH(S)Veff and X ∈ SH(S)proj . If for each fixed n the groups Hom(X, Ei [n]) stabilize as i tends to minus infinity, then the spectral sequence of the tower with E2 -term Hom(X, Q∗ [∗]) and target graded group Hom(X, E[∗]) converges strongly. Proof. Smashing the tower with the Spanier-Whitehead dual D(X) of X produces a Veff for a fixed integer n. Hence we may tower with terms D(X) ∧ Ei ∈ Σi+n T SH(S) assume that X is the sphere spectrum because smooth projective schemes of finite type over S are dualizable [23]. The spectral sequence obtained from the exact couple associated to the tower is strongly convergent due to Lemma 5.10. For the complex cobordism spectrum MU, fix an isomorphism MU∗ ∼ = Z[x1 , x2 , . . . ] where |xi | = i and consider the canonical map MU∗ → MGL∗ . Proposition 5.12: Over fields of characteristic zero there is a natural isomorphism from the quotient of MGL by the sequence (xi )i≥2 to kgl = f0 KGL. Proof. The orientation map MGL → KGL sends xi ∈ MGL2i,i to 0 in KGL2i,i for i ≥ 2. Hence there is a naturally induced map from the quotient of MGL by the sequence (xi )i≥2

24

to KGL. Since the quotient MGL/(xi )i≥2 is an effective spectrum we obtain the desired map to kgl. As shown in [42, Proposition 5.4] this map induces an isomorphism on all slices. (The proof employs the work of Hopkins-Morel on quotients of MGL.) For any X of SH(S)proj we may consider the spectral sequences obtained by taking homs into the respective slice filtrations of kgl and the quotient. Theorem 5.7 and Proposition 5.11 ensure that the spectral sequence for the quotient is strongly convergent. For kgl, strong convergence holds by [49, Proposition 5.5]. (Note that Conjecture 4 in [49] is proven in [27], cf. the introduction in loc. cit. for a discussion.) Our claim follows now by comparing the target graded groups of these spectral sequences. Corollary 5.13: Over fields of characteristic zero the connective K-theory spectrum kgl is very effective. Proof. Combine Theorem 5.7 and Proposition 5.12 with the fact that very effectiveness is preserved under homotopy colimits. Lemma 5.14: The motivic spectrum Σ∞ X+τ ∧Σ∞ P∞ kgl is very effective. + ∧n ∧ kgl Proof. For n ≥ 0, Corollary 5.13 shows the smash product Σ∞ X+τ ∧ (Σ∞ P∞ +) is very effective since SH(S)Veff is closed under smash products in SH(S) according to Lemma 5.6. When n varies, ∧n n 7→ Σ∞ X+τ ∧ (Σ∞ P∞ ∧ kgl +)

defines a simplicial object in motivic symmetric spectra. Its homotopy colimit is very kgl. effective and identifies with the smash product Σ∞ X+τ ∧Σ∞ P∞ + proj the full thick subcategory of the homotopy We denote by Ho(Σ∞ P∞ + − Mod) i ∞ ∞ ∞ category Ho(Σ∞ P∞ + − Mod) generated by the objects ΣT Σ P+ ∧ Σ X+ for X ∈ Sm projective and i ∈ Z. proj . Then there Lemma 5.15: Suppose Σ∞ X+τ is an object of Ho(Σ∞ P∞ + − Mod) exists an integer n ∈ Z such that HomΣ∞ P∞ (Σ∞ X+τ , kgl) lies in ΣnT SH(S)Veff . +

Proof. Note first that for every X ∈ Sm, i, j ∈ Z, there is an n1 ∈ Z such that Σi,j Σ∞ X+ ∈ ΣnT1 SH(S)Veff . For X projective we get that D(Σ∞ X+ ) ∈ ΣnT2 SH(S)Veff for some n2 ∈ Z by [23, Appendix] and we conclude that ∞ ∼ −i,−j D(Σ∞ X+ ) ∧ kgl (Σi,j Σ∞ P∞ HomΣ∞ P∞ + ∧ Σ X+ , kgl) = Σ +

25

lies in ΣnT3 SH(S)Veff for some n3 ∈ Z by Corollary 5.13. This shows the result for all of proj . The general case follows routinely by taking the generators of Ho(Σ∞ P∞ + − Mod) cones and direct summands. Theorem 5.16: Let S be a field of characteristic zero. Suppose Σ∞ X+ is compact, equivalently strongly dualizable, equivalently an object of SH(S)proj . Then the motivic twisted K-theory spectral sequence MZ∗ (Σ∞ X+ ) =⇒ KGLτ∗ (X ) in (16) is strongly convergent. Proof. The proof follows by reference to Proposition 5.11 in the case when X = S 0,q . Two assumptions need to be checked, i.e. Ei = Σ2i,i Σ∞ X+ ∧Σ∞ P∞ kgl ∈ ΣiP1 SH(S)Veff + and stability. Very effectiveness and Lemma 5.14 verify that the first assumption holds. Second, the stabilization condition is equivalent to the fact that for a fixed n, the group πn,q Qi (X τ ) 6= 0 for only finitely many i. The latter follows from the corresponding statement in the untwisted case because Σ∞ X+ is strongly dualizable. Namely, letting i ≪ 0 vary, the groups Hom(Σn,q D(Σ∞ X+ ), fi KGL) become isomorphic. Theorem 5.17: Let S be a field of characteristic zero. Suppose Σ∞ X+τ is compact in Ho(Σ∞ P∞ + − Mod), equivalently strongly dualizable. Then the motivic twisted K-theory spectral sequence MZ∗ (Σ∞ X+ ) =⇒ KGL∗τ (X ) in (17) is strongly convergent. Proof. We first note that the assumptions imply that Σ∞ X+ is strongly dualizable by Lemma 3.6. The proof proceeds now along the lines of the proof of Theorem 5.16 with a reference to Lemma 5.15 for very effectiveness. We end this section by discussing the closely related approach of the slice spectral sequence for KGLτ . Recall that the slices of any motivic spectrum fit into the slice tower constructed by Voevodsky in [48]. An identification of the zero-slice of KGLτ would in turn determine all the slices sn KGLτ by (2, 1)-periodicity, i.e. there is an isomorphism in the motivic stable homotopy category sn KGLτ ∼ = Σ2n,n s0 KGLτ .

26

This follows from the evident KGL-module structure on KGLτ and the Bott periodicity isomorphism β : Σ2,1 KGL → KGL furnishing the composite isomorphism Σ2,1 KGLτ → Σ2,1 KGLτ ∧ KGL → KGLτ ∧ Σ2,1 KGL → KGLτ ∧ KGL → KGLτ . The same comments apply to KGLτ . If the base scheme is a perfect field, then all of the slices sn KGLτ and sn KGLτ are in fact motives, i.e. modules over the integral motivic Eilenberg-MacLane spectrum MZ, cf. [27], [33], [38], [39], [50], [51]. However, except for example when X is the point and τ the trivial twist, the slice spectral sequences cannot coincide with the spectral sequence constructed earlier in this section. Indeed the corresponding filtration quotients are different because by weight considerations the smash product X ∧ MZ is not a zero slice in general.

6

Further problems and questions

We end the main body of the paper by discussing some problems and questions related to motivic twisted K-theory. A pressing question left open in the previous section is to identify the d1 -differentials in the spectral sequences. Problem 6.1: Express the d1 -differentials in the slice spectral sequence for KGLτ in terms of motivic Steenrod squares and the twist τ . Remark 6.2: The d3 -differentials in the Atiyah-Hirzebruch spectral sequence for twisted K-theory were identified by Atiyah and Segal in [6] as the difference between Sq3 and the twisting. In the same paper the higher differentials are determined in terms of Massey products. One may ask if also the higher differentials in the slice spectral sequence can be described in terms of Massey products. Problem 6.3: For twists τ and τ ′ construct products KGLτ ∧ KGLτ ′ → KGLτ +τ ′ and investigate its properties. In Remark 4.5 we noted that all the motivic twisted K-groups of the identity map of BBGm are trivial. More generally, if τ is any twisting of BBGm one may ask if the motivic τ -twisted K-groups are trivial. This is the content of the next problem asking when BBGm is a point for motivic twisted K-theory. 27

Problem 6.4: For which twists of BBGm is the associated motivic twisted K-theory trivial? Remark 6.5: The corresponding problem in topology has an affirmative solution for all twists by work of Anderson and Hodgkin [3]. By analogy, work on Problem 6.4 is likely to involve a computation of the KGL-homology of the motivic Eilenberg-MacLane spaces K(Z/n, 2) for n ≥ 1 any integer. Twisted equivariant K-theory for compact Lie groups is closely related to loop groups [14]. It is natural to ask for a generalization of our construction of motivic twisted Ktheory to an equivariant setting involving group schemes. Problem 6.6: Develop a theory of motivic twisted equivariant K-theory. The last problem we suggest is a very basic one. The construction of motivic twisted K-theory should generalize to other examples. We wish to single out hermitian K-theory as a closely related example of much interest. In this case we expect the twistings arise from homotopy classes of maps from X to the classifying space BBµ2 , i.e. elements of the second mod-2 motivic cohomology group MZ2,1 (X ; Z/2) of weight one. Problem 6.7: Develop a theory of motivic twisted hermitian K-theory.

7

Graded Adams Hopf algebroids

Recall that a Hopf algebroid is a cogroupoid object in the category of commutative rings [35, Appendix A1]. Let (A, Γ) be a Hopf algebroid. If the left unit ηL : A → Γ classifying the domain is flat, or equivalently the right unit ηR : A → Γ classifying the codomain is flat, then (A, Γ) is called a flat Hopf algebroid. If A → B is a ring map, we write B ⊗A Γ for the tensor product when Γ is given an A-module structure via ηL and Γ ⊗A B when Γ is given an A-module structure via ηR . An (A, Γ)-comodule comprises an A-module M together with a coassociative and unital map of left A-modules M → Γ ⊗A M (see e.g. [35, Appendix A1]). The category of (A, Γ)-comodules with the evident notion of a morphism is an abelian category provided Γ is a flat right A-module via ηR . Likewise, a graded Hopf algebroid is a cogroupoid object in the category of graded commutative rings [35, Appendix A1]. The notions of flat graded Hopf algebroids and comodules over a graded Hopf algebroid are defined exactly as in the ungraded setting. The examples of Hopf algebroids of main interest in stable homotopy theory are socalled “Adams Hopf algebroids.” In the graded setting we make the following definition:

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A graded Hopf algebroid (A, Γ) is called a graded Adams Hopf algebroid if Γ is the colimit of a filtered system of graded comodules which are finitely generated and projective as graded A-modules. Proposition 7.1: The pair (KGL∗ , KGL∗ KGL) is a flat graded Adams Hopf algebroid. Proof. We give two proofs of this result. Since homology commutes with sequential colimits we get that KGL∗ (P∞ ) is a filtered colimit of comodules which are finitely generated free KGL∗ -modules. Hence the same holds for KGL∗ KGL by using the Bott tower (1) for Σ∞ P∞ + as a model for KGL. For the second proof, recall the base change isomorphism in (5), KGL∗ KGL ∼ = KGL∗ ⊗KU∗ KU∗ KU. By the topological analogue of the first proof we see that (KU∗ , KU∗ KU) is a flat Adams Hopf algebroid. We conclude by pulling back the filtered colimit to the tensor product.

Proposition 7.2: Let (A, Γ) be a graded Hopf algebroid and A → B a graded ring map. Suppose B is a graded (A, Γ)-comodule algebra. (i) The pair (B, B ⊗A Γ) is a graded Hopf algebroid. (ii) If (A, Γ) is flat, then so is (B, B ⊗A Γ). (iii) If (A, Γ) is a graded Adams Hopf algebroid, then so is (B, B ⊗A Γ). Proof. For C be a graded (commutative) algebra, let X = Hom(A, C), M = Hom(Γ, C) and Y = Hom(B, C). Then (X, M ) is a groupoid and Y is a set over X equipped with an M -action. It is easily seen that the pair (Y, Y ×X M ) acquires the structure of a groupoid. This settles the first part. The second part follows by a standard base change argument, while the third point follows by pulling back the graded sub-comodules which are finitely generated projective as graded A-modules. The graded Hopf algebroid of primary interest in this paper is the following example. Proposition 7.3: The pair (KGL∗ (P∞ ), KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ )) is a flat graded Adams Hopf algebroid. 29

Proof. Note first that KGL∗ (P∞ ) has the structure of graded comodule algebra over (KU∗ , KU∗ KU). The result follows from Propositions 7.1 and 7.2. Remark 7.4: The left unit map denoted by ηKGL∗ (P∞ ) is determined by the composite map ∞ ∞ ∞ ∞ ∼ KGL ∧ Σ∞ P∞ + = KGL ∧ 1 ∧ Σ P+ → KGL ∧ KGL ∧ Σ P+ and the right unit map by ∞ ∞ ∞ ∞ ∼ KGL ∧ Σ∞ P∞ + = 1 ∧ KGL ∧ Σ P+ → KGL ∧ KGL ∧ Σ P+ .

(Here we make use of the unit map from the motivic sphere spectrum 1 to KGL.) We use the isomorphism ∞ ∼ KGL∗ (KGL ∧ Σ∞ P∞ + ) = KGL∗ KGL ⊗KGL∗ KGL∗ (P ).

Passing to KGL-homology under the left unit map yields a map KGL∗ (P∞ ) → KGL∗ KGL ⊗KGL∗ KGL∗ (P∞ ) displaying KGL∗ (P∞ ) as a comodule over KGL∗ KGL. The augmentation is determined by the multiplication on KGL via the map ∞ ∞ KGL ∧ KGL ∧ Σ∞ P∞ + → KGL ∧ Σ P+ .

The following is the graded version of the notion of Landweber exactness introduced by Hovey and Strickland in [22, Definition 2.1]. Definition 7.5: Suppose (A, Γ) is a flat graded Hopf algebroid. Then a graded ring map A → B is Landweber exact over (A, Γ) if the functor − ⊗A B from graded (A, Γ)comodules to graded B-modules is exact. By abuse of notation we let ηL denote the composite map ηL

A→Γ∼ = Γ ⊗A A → Γ ⊗A B. The next lemma is well known. For the convenience of the reader we shall sketch a proof since the result is employed in the proof of Theorem 4.3. Lemma 7.6: Suppose (A, Γ) is a flat graded Hopf algebroid. Then a graded ring map A → B is Landweber exact over (A, Γ) if and only if the map ηL : A → Γ ⊗A B is flat. 30

Proof. The only if implication holds because Γ ⊗A − preserves monomorphisms between graded A-modules. Conversely, for every graded A-comodule M , the graded coaction map M → Γ ⊗A M is a retract. Thus for a monomorphism of graded comodules M → N the map B ⊗A M → B ⊗A N is a retract of B ⊗A Γ ⊗A M → B ⊗A Γ ⊗A N . Remark 7.7: In the proof of Theorem 2.2 we could have worked with the Hopf algebroid (KU∗ (CP∞ ), KU∗ KU ⊗KU∗ KU∗ (CP∞ )) by restricting the comodule structure and using (ungraded) Landweber exactness. In this way one can bootstrap a proof of Theorem 2.2 more directly from [26] by using base change isomorphisms with no mention of graded Hopf algebroids. In the same spirit, we note there is an isomorphism KGLτ∗ (X ) ∼ = KGL∗ (X τ ) ⊗KU∗ (CP∞ ) KU∗ . Acknowledgements. We gratefully acknowledge hospitality and support from IMUB at the Universitat de Barcelona in the framework of the NILS mobility project. The second author benefitted from the hospitality of IAS in Princeton and TIFR in Mumbai during the preparation of this paper. Both authors are partially supported by the RCN 185335/V30.

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Department of Mathematics, University of Oslo, Norway. e-mail: [email protected] Department of Mathematics, University of Oslo, Norway. e-mail: [email protected]

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